In this series:

In part 1 of this blog series, we introduced the concept of inter-symbol interference (ISI) in OFDM systems and explained how to handle it. We also saw that ISI is usually created in multipath fading channels. In this second post on the subject, we will see that ISI can also come directly from the physical layer itself.

In Nutaq’s OFDM reference design, upsampling and downsampling finite impulse response (FIR) filters are used to match sampling rate of the digital-to-analog converters (DACs) and the physical layer. It's well-known that the best filter for minimizing ISI and converging towards an ideal or brick-wall filter is the raised-cosine filter (RRC). Depending on the roll-off factor, this filter can produce an excellent frequency response for upsampling/downsampling purposes:

Let’s look at the mathematical expression of the impulse response for the filter:

The impulse response of this filter has an infinite length. Digitally, we can’t create such a long impulse response without infinite resources, so a truncation of the ideal impulse response is inevitable which, of course, places us further away from our ideal brick-wall filter. One might intuitively use the highest possible order to get as near as possible to this ideal frequency response (that’s actually what I did the first time), but it’s important to note that the order of an FIR filter is directly linked to the number of taps in the impulse response. Since the filtering of a signal is basically the convolution of the signal with the impulse response of that filter, a bigger order means more taps and a longer impulse response.

Let’s look at an example. Here's the impulse response of an RRC FIR filter, order 20:

The effect of an impulse entering the filter at time 0 is, at its peak, approximately 260 ns later and ends up at 520 ns. In other words, there is 260 ns where the impulse still produces an effect at the output of the filter. Here is the same filter but with an order of 40:

We can easily see that the impulse response of this filter is twice as long as the previous one, thus creating a better quality frequency response.

But let’s suppose that the input of that filter is a series of two OFDM symbols with a cyclic prefix extension as shown in Part 1 of this blog series. What would happen? The answer: if the lengths of the cyclic prefix extension are smaller than the length of the impulse response of the RRC filter, ISI will be generated, without even going over the air!

The explanation is simple. The output of the IFFT in an OFDM system is a series of time samples that can be interpreted as a train of impulses of different amplitudes. So, if no cyclic prefix is used or if the length of the cyclic prefix is smaller than the actual impulse response of the RRC filter, there will be ISI, as the latest time samples of one OFDM symbol can overlap the time samples of the next one.

## Conclusion

In this two-part series, we introduced the concept of ISI and also demonstrated how ISI can come from elsewhere besides the over-the-air channel. By being aware of the different sources of ISI, one can save a lot of debugging time.